Abstract

We theoretically study the role of dispersion in propagation of rotating beams in left-handed materials (LHMs). By modeling the rotating beam as a superposition of two rotating Laguerre-Gaussian beams with opposite chirality, same magnitude and different frequencies, we demonstrate that the rotation property of the rotating beam in LHM is significantly dependent on the sign and strength of dispersion: In the normal dispersion region, the direction of transverse energy flow is reversed compared to the vacuum, due to the negative refractive index of LHM, while in the anomalous dispersion region it may be parallel or antiparallel to that in the case of vacuum, depending on the strength of dispersion. In addition, we find that the angular momentum density can be parallel or antiparallel to the transverse energy flow in LHM, while the angular momentum flow is always opposite to the transverse energy flow.

© 2009 Optical Society of America

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  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  4. L. Allen, M. J. Padgett, and M. Babiker, "The orbit angular momentum of light," Prog. Opt. 39, 291-372 (1999).
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  5. M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
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  6. J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
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  7. C. N. Alexeyev and M. A. Yavorsky, "Angular momentum of rotating paraxial light beams," J. Opt. A: Pure Appl. Opt. 7, 416-421 (2005).
    [CrossRef]
  8. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Commun. 249, 367-378 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  14. M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles," Phys. Rev. A 54, 1593-1596 (1996).
    [CrossRef] [PubMed]
  15. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner," Opt. Lett. 22, 52-54 (1997).
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  16. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
    [CrossRef] [PubMed]
  17. M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
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  18. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. H. Luo, Z. Ren, W. Shu, and S. C. Wen, "Reversed propagation dynamics of Laguerre-Gaussian beams in lefthanded materials," Phys. Rev. A 77, 023812 (2008).
    [CrossRef]
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    [CrossRef]
  33. R. Loudon, L. Allen, and D. F. Nelson, "Propagation of electromagnetic energy and momentum through an absorbing dielectric," Phys. Rev. E 55, 1071-1085 (1997).
    [CrossRef]

2008 (2)

H. Luo, Z. Ren, W. Shu, and S. C. Wen, "Reversed propagation dynamics of Laguerre-Gaussian beams in lefthanded materials," Phys. Rev. A 77, 023812 (2008).
[CrossRef]

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

2007 (2)

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

2006 (4)

G. Nienhuis, "Polychromatic and rotating beams of light," J. Phys. B: At.Mol. Opt. Phys. 39, S529-S544 (2006).
[CrossRef]

H. Luo,W. Shu, F. Li, and Z. Ren, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

A. Ya. Bekshaev and M. S. Soskin, "Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons," Opt. Lett. 31, 2199-2201 (2006).
[CrossRef] [PubMed]

ChristianH. J. Schmitz, Kai Uhrig, Joachim P. Spatz and, J. E. Curtis, "Tuning the orbital angular momentum in optical vortex beams," Opt. Express 14, 6604-6612 (2006).
[CrossRef] [PubMed]

2005 (2)

C. N. Alexeyev and M. A. Yavorsky, "Angular momentum of rotating paraxial light beams," J. Opt. A: Pure Appl. Opt. 7, 416-421 (2005).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Commun. 249, 367-378 (2005).
[CrossRef]

2003 (2)

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003).
[CrossRef]

2002 (2)

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

2001 (4)

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-253 (2001).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001)
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

2000 (1)

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67-71 (2000).
[CrossRef]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbit angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

1997 (2)

R. Loudon, L. Allen, and D. F. Nelson, "Propagation of electromagnetic energy and momentum through an absorbing dielectric," Phys. Rev. E 55, 1071-1085 (1997).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

1996 (1)

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles," Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

1995 (2)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1981 (1)

J. F. Nye, "The motion and structure of dislocations in wavefronts," Proc. R. Soc. Lond. A. 378, 219-239 (1981).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Alexeyev, C. N.

C. N. Alexeyev and M. A. Yavorsky, "Angular momentum of rotating paraxial light beams," J. Opt. A: Pure Appl. Opt. 7, 416-421 (2005).
[CrossRef]

Allen, L.

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67-71 (2000).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbit angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

R. Loudon, L. Allen, and D. F. Nelson, "Propagation of electromagnetic energy and momentum through an absorbing dielectric," Phys. Rev. E 55, 1071-1085 (1997).
[CrossRef]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Arlt, J.

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, "The orbit angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Bekshaev, A. Ya.

A. Ya. Bekshaev and M. S. Soskin, "Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons," Opt. Lett. 31, 2199-2201 (2006).
[CrossRef] [PubMed]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Commun. 249, 367-378 (2005).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Christian,

Curtis, J. E.

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

Dholakia, K.

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

Enger, J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles," Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

Fan, D.

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

Friese, M. E. J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles," Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Galajda, P.

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-253 (2001).
[CrossRef]

Grier, D. G.

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

Grzegorczyk, T. M.

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles," Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Kemp, B. A.

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

Kong, J. A.

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

Li, F.

H. Luo,W. Shu, F. Li, and Z. Ren, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

Loudon, R.

R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003).
[CrossRef]

R. Loudon, L. Allen, and D. F. Nelson, "Propagation of electromagnetic energy and momentum through an absorbing dielectric," Phys. Rev. E 55, 1071-1085 (1997).
[CrossRef]

Luo, H.

H. Luo, Z. Ren, W. Shu, and S. C. Wen, "Reversed propagation dynamics of Laguerre-Gaussian beams in lefthanded materials," Phys. Rev. A 77, 023812 (2008).
[CrossRef]

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

H. Luo,W. Shu, F. Li, and Z. Ren, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Nelson, D. F.

R. Loudon, L. Allen, and D. F. Nelson, "Propagation of electromagnetic energy and momentum through an absorbing dielectric," Phys. Rev. E 55, 1071-1085 (1997).
[CrossRef]

Nienhuis, G.

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

G. Nienhuis, "Polychromatic and rotating beams of light," J. Phys. B: At.Mol. Opt. Phys. 39, S529-S544 (2006).
[CrossRef]

Nye, J. F.

J. F. Nye, "The motion and structure of dislocations in wavefronts," Proc. R. Soc. Lond. A. 378, 219-239 (1981).
[CrossRef]

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Ormos, P.

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-253 (2001).
[CrossRef]

Padgett, M. J.

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67-71 (2000).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbit angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner," Opt. Lett. 22, 52-54 (1997).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995).
[CrossRef]

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Ren, Z.

H. Luo, Z. Ren, W. Shu, and S. C. Wen, "Reversed propagation dynamics of Laguerre-Gaussian beams in lefthanded materials," Phys. Rev. A 77, 023812 (2008).
[CrossRef]

H. Luo,W. Shu, F. Li, and Z. Ren, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles," Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001)
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001)
[CrossRef] [PubMed]

Shu, W.

H. Luo, Z. Ren, W. Shu, and S. C. Wen, "Reversed propagation dynamics of Laguerre-Gaussian beams in lefthanded materials," Phys. Rev. A 77, 023812 (2008).
[CrossRef]

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

H. Luo,W. Shu, F. Li, and Z. Ren, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

Sibbett, W.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

Simpson, N. B.

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001)
[CrossRef] [PubMed]

Soskin, M. S.

A. Ya. Bekshaev and M. S. Soskin, "Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons," Opt. Lett. 31, 2199-2201 (2006).
[CrossRef] [PubMed]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Commun. 249, 367-378 (2005).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Tang, Z.

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

van Enk, S. J.

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Commun. 249, 367-378 (2005).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Volke-Sepulveda, K.

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

Wen, S. C.

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

H. Luo, Z. Ren, W. Shu, and S. C. Wen, "Reversed propagation dynamics of Laguerre-Gaussian beams in lefthanded materials," Phys. Rev. A 77, 023812 (2008).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Yavorsky, M. A.

C. N. Alexeyev and M. A. Yavorsky, "Angular momentum of rotating paraxial light beams," J. Opt. A: Pure Appl. Opt. 7, 416-421 (2005).
[CrossRef]

Zou, Y.

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-253 (2001).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

C. N. Alexeyev and M. A. Yavorsky, "Angular momentum of rotating paraxial light beams," J. Opt. A: Pure Appl. Opt. 7, 416-421 (2005).
[CrossRef]

J. Phys. B: At.Mol. Opt. Phys. (1)

G. Nienhuis, "Polychromatic and rotating beams of light," J. Phys. B: At.Mol. Opt. Phys. 39, S529-S544 (2006).
[CrossRef]

Opt. Commun. (5)

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, "Angular momentum of a rotating light beam," Opt. Commun. 249, 367-378 (2005).
[CrossRef]

H. Luo,W. Shu, F. Li, and Z. Ren, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995).
[CrossRef]

L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67-71 (2000).
[CrossRef]

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, "Revolving interference patterns for the rotation of optically trapped particles," Opt. Commun. 201, 21-29 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (7)

B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, "Reversal of wave momentum in isotropic left-handed media," Phys. Rev. A 75, 053810 (2007).
[CrossRef]

R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003).
[CrossRef]

H. Luo, Z. Ren, W. Shu, and S. C. Wen, "Reversed propagation dynamics of Laguerre-Gaussian beams in lefthanded materials," Phys. Rev. A 77, 023812 (2008).
[CrossRef]

H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, "Rotational Doppler effect in left-handed materials," Phys. Rev. A 78, 033805 (2008).
[CrossRef]

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles," Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

R. Loudon, L. Allen, and D. F. Nelson, "Propagation of electromagnetic energy and momentum through an absorbing dielectric," Phys. Rev. E 55, 1071-1085 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

J. E. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (1)

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Proc. R. Soc. Lond. A. (1)

J. F. Nye, "The motion and structure of dislocations in wavefronts," Proc. R. Soc. Lond. A. 378, 219-239 (1981).
[CrossRef]

Prog. Opt. (2)

L. Allen, M. J. Padgett, and M. Babiker, "The orbit angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001).
[CrossRef]

Science (3)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia1, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef] [PubMed]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of Three-Dimensional optically trapped structures," Science 296, 1101-1103 (2002).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001)
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (4)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, MA, 2005).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, CA, 1980).

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Figures (4)

Fig. 1.
Fig. 1.

Evolution of the interference pattern of model spiral beams viewed against the beam propagation direction in the waist plane. The white dot indicates the rotation of the pattern due to the azimuthal index of two-term superposition of LG beams. Here, Ω is the angular velocity of interference pattern, T = 2(l + - l -)πω is the period of rotation. (a)-(d) superposition LG0,+1 +LG0,-1: we know that Ω > 0 and the interference pattern exhibits anticlockwise rotation. (a’)-(d’) superposition LG1,-1 +LG1,+1: we get that Ω > 0 and the interference pattern exhibits clockwise rotation.

Fig. 2.
Fig. 2.

Numerically computed field intensity distribution and transverse energy flow (green arrows) of RHG beams in vacuum (left) and in the normal dispersion region of the LHM (right), respectively, at z = 0. (a) and (a’): superposition LG0,+1 + LG0,-1, the transverse Poynting vector exhibits anticlockwise and clockwise spiral in vacuum and in the normal dispersion region of the LHM, respectively. (b) and (b’): superposition LG1,-1 + LG1,+1, the transverse Poynting vector presents clockwise and anticlockwise spiral in vacuum and in the normal dispersion region of the LHM, respectively.

Fig. 3.
Fig. 3.

Numerically computed field intensity distribution and transverse energy flow (green arrows) of RHG beams in the anomalous dispersion region of LHMs at z = 0. (a) and (a’): superposition LG0+1 + LG0-1, the transverse Poynting vector exhibits clockwise and anticlockwise spiral since |(∂η/∂ω)| ω=ω 0 | > 2|η(ω 0)|/ω 0 and |(∂η/∂ω)| ω=ω 0 | < 2|η(ω 0)|/ω 0 in (a) and (a’), respectively. (b) and (b’): superposition LG1, -1 + LG1, +1, the transverse Poynting vector presents anticlockwise and clockwise spiral since | (∂η/∂ω)| ω=ω 0 | < 2|η(ω 0)|/ω 0 and | (∂η/∂ω)| ω=ω 0 | > 2|η(ω 0)|/ω 0 in (b) and (b’), respectively.

Fig. 4.
Fig. 4.

Numerically computed field intensity distribution and angular momentum flow (green arrows) of RHG beams in the LHM for (∂η/∂ω)| ω=ω 0 | > 2η(ω 0)/ω 0/ω 0 (left) and (∂η/∂ω)| ω=ω 0 < 2η(ω 0)/(ω 0) (right). White arrow shows the direction of S φ. (a) and (a’): superposition LG0, +1 +LG0,-1. (b) and (b’): superposition LG1, -1 +LG1, +1. Note that the angular momentum flow is always opposite to the transverse energy flow.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

u ˜ ( k ) = 0 drr J l ( kr ) u ( r , 0 ) ,
u ( r , z ) = 0 dkk exp ( i k 2 z 2 n R , L k 0 ) J l ( kr ) u ˜ ( k ) .
u ( r , 0 ) = c pl w 0 ( r w 0 ) l L p l ( r 2 w 0 2 ) exp ( r 2 2 w 0 2 + ik r 2 2 R 0 ) exp ( ilφ ) ,
u ( r , z ) = c pl w ( z ) [ r w ( z ) ] l L p l [ r 2 w 2 ( z ) ] exp [ r 2 2 w 2 ( z ) + i k 0 n R r 2 2 R ( z ) ]
× exp ( ilφ ) exp [ i ( 2 p + l + 1 ) ψ ( z ) ] ,
u ( r , z ) = c pl w ( z ) [ r w ( z ) ] l L p l [ r 2 w 2 ( z ) ] exp [ r 2 2 w 2 ( z ) + i k 0 n L r 2 2 R ( z ) ]
× exp ( ilφ ) exp [ i ( 2 p + l + 1 ) ψ ( z ) ] ,
E ( r , z ) = iωu e x c n u x e z ,
B ( r , z ) = iku e y u y e z ,
ε ( ω ) = ε 0 ( 1 ω ep 2 ω e 0 2 ω 2 ω e 0 2 + γ e ) ,
μ ( ω ) = μ 0 ( 1 ω mp 2 ω m 0 2 ω 2 ω m 0 2 + γ m ) ,
E = E + + E , H = H + + H ,
E ± ( r , z , t ) = 1 2 { E ± ( r , z ) exp [ i ( k ± z ω ± t ) ] + E ± * ( r , z ) exp [ i ( k ± z ω ± t ) ] } ,
H ± ( r , z , t ) = 1 2 { H ± ( r , z ) exp [ i ( k ± z ω ± t ) ] + H ± * ( r , z ) exp [ i ( k ± z ω ± t ) ] } ,
S = ( E + + E ) × ( H + + H ) ,
S = S z + S .
S z = 1 2 c { ω + 2 u + 2 exp ( 2 υ + z ) + ω 2 u 2 exp ( 2 υ z )
+ 2 ω + ω u + u exp [ ( υ + + υ ) z ] cos ( γ ) } e z ,
γ = ( l + l ) φ Δ ωt + ( τ + τ ) z + c ψ ( z ) c + ψ ( z ) ,
S z = 1 2 c { ω + 2 u + 2 + ω 2 u 2 + 2 ω + ω u + u cos [ ( l + + l ) φ Δ ωt ] } e z .
S x = i 4 [ ω + η + ( u + u + * x u + * u + * x ) + ω η ( u u * x u * u x )
+ ( ω + η u + u * x ω η + u * u + x ) exp ( i γ )
+ ( ω η + u u + * x ω + η u + * u x ) exp ( i γ ) ] e x ,
S y = i 4 [ ω + η + ( u + u + * y u + * u + y ) + ω η ( u u * y u * u y )
+ ( ω + η u + u * y ω η + u * u + y ) exp ( i γ )
+ ( ω η + u u + * y ω + η u + * u y ) exp ( i γ ) ] e y .
S = ω + η + ( u + u + * u + * u + ) + ω η ( u u * u * u ) ,
S φ = { ω + η + exp [ υ + r 2 R ( z ) ] ω η exp [ υ r 2 R ( z ) ] } l 2 u ( r , 0 ) 2 r e φ ,
S r = { ω + τ + η + exp [ υ + r 2 R ( z ) ] ω τ η exp [ υ r 2 R ( z ) ] } u ( r , 0 ) 2 r 2 R ( z ) e r .
S φ = ( ω + η + ω η ) l 2 u r e φ ,
S φ = [ Δω η ( ω 0 ) ω 0 2 η 2 ( ω 0 ) η ω ω = ω 0 Δω + 1 8 ( 1 η ( ω ) ) ω = ω 0 Δ ω 3 + ] l 2 r u 2 e φ ,
S φ = [ Δω η ( ω 0 ) + ω 0 2 η 2 ( ω 0 ) η ω ω = ω 0 Δω ] l 2 r u 2 e φ .
· T + G t = F ,
G = 1 2 Re [ εμ E × H * + k 2 ( ε ω E 2 + μ ω H 2 ) ] ,
T = 1 2 Re [ ( E * + H * ) I ( DE * + BH * ) ] ,
F = 1 2 Re [ ρ e E * + J × B * + ρ m H * + M × D * ] ,
G = 1 2 Re [ n ( n + ω n ω ) E 2 ] e s ,
v e = T G = η 2 + κ 2 + 2 ( ω 0 ηκ / γ e ) e s .

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